User blog:Rgetar/New 9 lists of ordinals (total 5439 ordinals)
This is annex to Fixed program blog with lists of ordinals, generated using Ordinal Explorer v3.1. March 24, 2019 I published 9 lists of ordinals, generated using v3.0 of my program. Bubby3 (thank you Bubby3!) wrote in a comment that I skip many ordinals like [[MM]M][MM] = psi(W_2+psi_1(W_2)) = psi(e_(W+1)*2) and [[[[MM]]M][MM]] = psi(psi_I(0)*2). In that blog I wrote that limit of these lists is "Rathjen ordinal" (#1201 in Scorcher007's list of 2000 ordinals), but it meant that this limit is lesser. I think that Bubby3 meant [[[MM]M]MM], not [[MM]M][MM]. I think that :[[MM]M][MM] = [Ω2Ω]Ω2 is non-standard form of :[MM][MM] = Ω2Ω2 At least my program generates almost the same fundamental sequences for these strings (using "modified" fs algorythm). For [[MM]M][MM]: :[][MM] = BHO + 1 :[[]M][MM] = εBHO + ω :[[[]M]M][MM] = φ(ω, BHO + 1) :[[[[]M]M]M][MM] = φ(1 @ ω, BHO + 1 @ 0) :[[[[[]M]M]M]M][MM] :... For [MM][MM]: :empty string = 0 :[MM] = BHO :[][MM] = BHO + 1 :[[]M][MM]= εBHO + ω :[[[]M]M][MM] = φ(ω, BHO + 1) :[[[[]M]M]M][MM] = φ(1 @ ω, BHO + 1 @ 0) :[[[[[]M]M]M]M][MM] :... That is fundamental sequences are the same, but for [MM][MM] it contains two additional elements in the beginning. So, I think that [[MM]M][MM] = [MM][MM], and using psi it can be written as psi(e_(W+1) + 1) (I am not sure, since I do not know psi OCF very well). Then :[[]MM] = [[]Ω2] = + 1 = psi(e_(W+1) + ω) :[MMM] = [ΩΩ2] = + Ω = psi(e_(W+1) + Ω) :[[[MM]M]MM] = [[Ω2Ω]Ω2] = + εΩ + 1 = psi(W_2+psi_1(W_2)) = psi(e_(W+1)2) [[[[MM]]M][MM]] = [[IM]I] = [+ II] = psi(psi_I(0)*2) If these new lists both strings ([[[MM]M]MM] and [[[[MM]]M][MM]]) are present. [[MM]M][MM] is not present. I think it is because it is non-standard form of [MM][MM], and [MM][MM] is present. But I am still not sure that there are no other missed ordinals. So, new lists. Theirs limit (L) is still perhaps "Rathjen ordinal". In lists with additional fs elements I inserted 5 additional fs elements of each limit ordinal. Sizes of "clean" lists (without additional fs elements): :Single expansion: 4 ordinals :Double expansion: 12 ordinals :Triple expansion: 56 ordinals :Quadruple expansion: 353 ordinals :Pentuple expansion: 2833 ordinals :Hexuple expansion (not published here): 29331 ordinals :Heptuple expansion (not published here): 406327 ordinals Sizes of lists with additional fs elements: :Single expansion: 14 ordinals :Double expansion: 52 ordinals :Triple expansion: 276 ordinals :Quadruple expansion: 1838 ordinals :Pentuple expansion (not published here): 15233 ordinals :Hexuple expansion (not published here): 161821 ordinals :Heptuple expansion (not published here): 2291307 ordinals Single expansion Clean [] M L Total 4 ordinals. With additional fs elements [] [[]] [] [[]] [[[]]] [[[[]]]] M [[MM]] [[[MM]M]] [[[[MM]M]M]] [[[[[MM]M]M]M]] [[[[[[MM]M]M]M]M]] L Total 14 ordinals. Double expansion Clean [] [][] [[]] M []M MM [[]M] [MM] [[[]M]] [[MM]] L Total 12 ordinals. With additional fs elements [] [][] [][][] [][][][] [][][][][] [][][][][][] [][][][][][][] [[]] [] [[]] [[[]]] [[[[]]]] [[[[[]]]]] M []M [[]M]M [[[]M]M]M [[[[]M]M]M]M [[[[[]M]M]M]M]M [[[[[[]M]M]M]M]M]M MM MMM MMMM MMMMM MMMMMM MMMMMMM [[]M] [[[]M]M] [[[[]M]M]M] [[[[[]M]M]M]M] [[[[[[]M]M]M]M]M] [[[[[[[]M]M]M]M]M]M] [MM] [MMM] [MMMM] [MMMMM] [MMMMMM] [MMMMMMM] [[[]M]] [[[[[]M]]M]] [[[[[[[]M]]M]]M]] [[[[[[[[[]M]]M]]M]]M]] [[[[[[[[[[[]M]]M]]M]]M]]M]] [[[[[[[[[[[[[]M]]M]]M]]M]]M]]M]] [[MM]] [[[MM]M]] [[[[MM]M]M]] [[[[[MM]M]M]M]] [[[[[[MM]M]M]M]M]] [[[[[[[MM]M]M]M]M]M]] L Total 52 ordinals. Triple expansion Clean [] [][] [][][] [[]] [][[]] [[]][[]] [[][]] [] M []M [][]M [[]]M [M]M [[]M]M MM []MM MMM [[]M] [][[]M] M[[]M] [[]M][[]M] [[][]M] [M] [[M]M] [[[[]M]]M] [MM] [[[]M]M] [MM] [][MM] [MM][MM] [[]MM] [MMM] [[[]M]] [][[[]M]] [[[]M]][[[]M]] [[][[]M]] [M[[]M]] [[[]M][[]M]] [[[][]M]] [[M]] [[[[[[]M]]]M]] [[MM]] [[[[[]M]]M]] [[MM]] [][[MM]] [[MM]][[MM]] [[][MM]] [M[MM]] [[[]M][MM]] [[MM][MM]] [[[]MM]] [[MMM]] [[[[]M]M]] [[[MM]M]] L Total 56 ordinals. With additional fs elements [] [][] [][][] [][][][] [][][][][] [][][][][][] [][][][][][][] [][][][][][][][] [[]] [][[]] [][][[]] [][][][[]] [][][][][[]] [][][][][][[]] [][][][][][][[]] [[]][[]] [[]][[]][[]] [[]][[]][[]][[]] [[]][[]][[]][[]][[]] [[]][[]][[]][[]][[]][[]] [[]][[]][[]][[]][[]][[]][[]] [[][]] [[][][]] [[][][][]] [[][][][][]] [[][][][][][]] [[][][][][][][]] [] [[]] [[[]]] [[[[]]]] [[[[[]]]]] [[[[[[]]]]]] M []M [][]M [][][]M [][][][]M [][][][][]M [][][][][][]M [][][][][][][]M [[]]M []M [[]]M [[[]]]M [[[[]]]]M [[[[[]]]]]M [M]M [M][M]M [M][M][M]M [M][M][M][M]M [M][M][M][M][M]M [M][M][M][M][M][M]M [[]M]M [[[]M]M]M [[[[]M]M]M]M [[[[[]M]M]M]M]M [[[[[[]M]M]M]M]M]M [[[[[[[]M]M]M]M]M]M]M MM []MM [[]MM]MM [[[]MM]MM]MM [[[[]MM]MM]MM]MM [[[[[]MM]MM]MM]MM]MM [[[[[[]MM]MM]MM]MM]MM]MM MMM MMMM MMMMM MMMMMM MMMMMMM MMMMMMMM [[]M] [][[]M] [[][[]M]][[]M] [[[][[]M]][[]M]][[]M] [[[[][[]M]][[]M]][[]M]][[]M] [[[[[][[]M]][[]M]][[]M]][[]M]][[]M] [[[[[[][[]M]][[]M]][[]M]][[]M]][[]M]][[]M] M[[]M] MM[[]M] MMM[[]M] MMMM[[]M] MMMMM[[]M] MMMMMM[[]M] [[]M][[]M] [[]M][[]M][[]M] [[]M][[]M][[]M][[]M] [[]M][[]M][[]M][[]M][[]M] [[]M][[]M][[]M][[]M][[]M][[]M] [[]M][[]M][[]M][[]M][[]M][[]M][[]M] [[][]M] [[][][]M] [[][][][]M] [[][][][][]M] [[][][][][][]M] [[][][][][][][]M] [M] [[]M] [[[]]M] [[[[]]]M] [[[[[]]]]M] [[[[[[]]]]]M] [[M]M] [[MM]M] [[MMM]M] [[MMMM]M] [[MMMMM]M] [[MMMMMM]M] [[[[]M]]M] [[[[[[]M]]M]]M] [[[[[[[[]M]]M]]M]]M] [[[[[[[[[[]M]]M]]M]]M]]M] [[[[[[[[[[[[]M]]M]]M]]M]]M]]M] [[[[[[[[[[[[[[]M]]M]]M]]M]]M]]M]]M] [MM] [MMM] [MMMM] [MMMMM] [MMMMMM] [MMMMMMM] [[[]M]M] [[[[]M]M]M] [[[[[]M]M]M]M] [[[[[[]M]M]M]M]M] [[[[[[[]M]M]M]M]M]M] [[[[[[[[]M]M]M]M]M]M]M] [MM] [][MM] [[]M][MM] [[[]M]M][MM] [[[[]M]M]M][MM] [[[[[]M]M]M]M][MM] [[[[[[]M]M]M]M]M][MM] [MM][MM] [MM][MM][MM] [MM][MM][MM][MM] [MM][MM][MM][MM][MM] [MM][MM][MM][MM][MM][MM] [MM][MM][MM][MM][MM][MM][MM] [[]MM] [[[]MM]MM] [[[[]MM]MM]MM] [[[[[]MM]MM]MM]MM] [[[[[[]MM]MM]MM]MM]MM] [[[[[[[]MM]MM]MM]MM]MM]MM] [MMM] [MMMM] [MMMMM] [MMMMMM] [MMMMMMM] [MMMMMMMM] [[[]M]] [][[[]M]] [][[[]M]] M[[[]M]] [MM][[[]M]] [MMM][[[]M]] [MMMM][[[]M]] [[[]M]][[[]M]] [[[]M]][[[]M]][[[]M]] [[[]M]][[[]M]][[[]M]][[[]M]] [[[]M]][[[]M]][[[]M]][[[]M]][[[]M]] [[[]M]][[[]M]][[[]M]][[[]M]][[[]M]][[[]M]] [[[]M]][[[]M]][[[]M]][[[]M]][[[]M]][[[]M]][[[]M]] [[][[]M]] [[[][[]M]][[]M]] [[[[][[]M]][[]M]][[]M]] [[[[[][[]M]][[]M]][[]M]][[]M]] [[[[[[][[]M]][[]M]][[]M]][[]M]][[]M]] [[[[[[[][[]M]][[]M]][[]M]][[]M]][[]M]][[]M]] [M[[]M]] [MM[[]M]] [MMM[[]M]] [MMMM[[]M]] [MMMMM[[]M]] [MMMMMM[[]M]] [[[]M][[]M]] [[[]M][[]M][[]M]] [[[]M][[]M][[]M][[]M]] [[[]M][[]M][[]M][[]M][[]M]] [[[]M][[]M][[]M][[]M][[]M][[]M]] [[[]M][[]M][[]M][[]M][[]M][[]M][[]M]] [[[][]M]] [[[][][]M]] [[[][][][]M]] [[[][][][][]M]] [[[][][][][][]M]] [[[][][][][][][]M]] [[M]] [[M]] [[[M]M]] [[[[MM]]M]] [[[[MMM]]M]] [[[[MMMM]]M]] [[[[[[]M]]]M]] [[[[[[[[[]M]]]M]]]M]] [[[[[[[[[[[[]M]]]M]]]M]]]M]] [[[[[[[[[[[[[[[]M]]]M]]]M]]]M]]]M]] [[[[[[[[[[[[[[[[[[]M]]]M]]]M]]]M]]]M]]]M]] [[[[[[[[[[[[[[[[[[[[[]M]]]M]]]M]]]M]]]M]]]M]]]M]] [[MM]] [[[MM]M]] [[[MMM]M]] [[[MMMM]M]] [[[MMMMM]M]] [[[MMMMMM]M]] [[[[[]M]]M]] [[[[[[[]M]]M]]M]] [[[[[[[[[]M]]M]]M]]M]] [[[[[[[[[[[]M]]M]]M]]M]]M]] [[[[[[[[[[[[[]M]]M]]M]]M]]M]]M]] [[[[[[[[[[[[[[[]M]]M]]M]]M]]M]]M]]M]] [[MM]] [][[MM]] [[[]M]][[MM]] [[[[[]M]]M]][[MM]] [[[[[[[]M]]M]]M]][[MM]] [[[[[[[[[]M]]M]]M]]M]][[MM]] [[[[[[[[[[[]M]]M]]M]]M]]M]][[MM]] [[MM]][[MM]] [[MM]][[MM]][[MM]] [[MM]][[MM]][[MM]][[MM]] [[MM]][[MM]][[MM]][[MM]][[MM]] [[MM]][[MM]][[MM]][[MM]][[MM]][[MM]] [[MM]][[MM]][[MM]][[MM]][[MM]][[MM]][[MM]] [[][MM]] [[[][MM]][MM]] [[[[][MM]][MM]][MM]] [[[[[][MM]][MM]][MM]][MM]] [[[[[[][MM]][MM]][MM]][MM]][MM]] [[[[[[[][MM]][MM]][MM]][MM]][MM]][MM]] [M[MM]] [MM[MM]] [MMM[MM]] [MMMM[MM]] [MMMMM[MM]] [MMMMMM[MM]] [[[]M][MM]] [[[[[]M][MM]]M][MM]] [[[[[[[]M][MM]]M][MM]]M][MM]] [[[[[[[[[]M][MM]]M][MM]]M][MM]]M][MM]] [[[[[[[[[[[]M][MM]]M][MM]]M][MM]]M][MM]]M][MM]] [[[[[[[[[[[[[]M][MM]]M][MM]]M][MM]]M][MM]]M][MM]]M][MM]] [[MM][MM]] [[MM][MM][MM]] [[MM][MM][MM][MM]] [[MM][MM][MM][MM][MM]] [[MM][MM][MM][MM][MM][MM]] [[MM][MM][MM][MM][MM][MM][MM]] [[[]MM]] [[[[[]MM]]MM]] [[[[[[[]MM]]MM]]MM]] [[[[[[[[[]MM]]MM]]MM]]MM]] [[[[[[[[[[[]MM]]MM]]MM]]MM]]MM]] [[[[[[[[[[[[[]MM]]MM]]MM]]MM]]MM]]MM]] [[MMM]] [[MMMM]] [[MMMMM]] [[MMMMMM]] [[MMMMMMM]] [[MMMMMMMM]] [[[[]M]M]] [[[[[[[]M]M]]M]M]] [[[[[[[[[[]M]M]]M]M]]M]M]] [[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]] [[[[[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]]M]M]] [[[[[[[[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]]M]M]]M]M]] [[[MM]M]] [[[[MM]M]M]] [[[[[MM]M]M]M]] [[[[[[MM]M]M]M]M]] [[[[[[[MM]M]M]M]M]M]] [[[[[[[[MM]M]M]M]M]M]M]] L Total 276 ordinals. Quadruple expansion Clean [] [][] [][][] [][][][] [[]] [][[]] [][][[]] [[]][[]] [][[]][[]] [[]][[]][[]] [[][]] [][[][]] [[]][[][]] [[][]][[][]] [[][][]] [] [][] [][] [[][[]]] [[[]]] [[[][]]] [[]] M []M [][]M [][][]M [[]]M [][[]]M [[]][[]]M [[][]]M []M [M]M [][M]M [M][M]M [[]M]M [][[]M]M [M][[]M]M [[]M][[]M]M [[][]M]M [M]M [[M]M]M [[[]M]M]M MM []MM [][]MM [[]]MM [M]MM [[]M]MM [MM]MM [[]MM]MM MMM []MMM MMMM [[]M] [][[]M] [][][[]M] [[]][[]M] [M][[]M] [[[]M]][[]M] [[][[]M]][[]M] M[[]M] []M[[]M] MM[[]M] [[]M][[]M] [][[]M][[]M] M[[]M][[]M] [[]M][[]M][[]M] [[][]M] [][[][]M] M[[][]M] [[]M][[][]M] [[][]M][[][]M] [[][][]M] [M] [][M] [M][M] [[][[]]M] [[[]]M] [[[][]]M] [[]M] [[M]M] [][[M]M] M[[M]M] [[]M][[M]M] [[M]M][[M]M] [[][M]M] [[M][M]M] [[[]M]M] [[MM]M] [[[[]M]]M] [][[[[]M]]M] M[[[[]M]]M] [[]M][[[[]M]]M] [[[[]M]]M][[[[]M]]M] [[][[[]M]]M] [[[[]M]][[[]M]]M] [[[][[]M]]M] [[M[[]M]]M] [[[[]M][[]M]]M] [[[[][]M]]M] [[[M]]M] [[[[M]M]]M] [[[[[[]M]]M]]M] [MM] [][MM] M[MM] [[]M][MM] [MM][MM] [[]MM] [MMM] [[[]M]M] [][[[]M]M] [[[]M]M][[[]M]M] [[][[]M]M] [M[[]M]M] [[[]M][[]M]M] [[[][]M]M] [[M]M] [[[[[[]M]M]]M]M] [[MM]M] [[[[]M]M]M] [MM] [][MM] [][][MM] [[]][MM] [[MM]][MM] [[][MM]][MM] M[MM] [[]M][MM] [MM][MM] [][MM][MM] [MM][MM][MM] [[]MM] [][[]MM] [MM][[]MM] [[]MM][[]MM] [[][]MM] [MM] [[[MM]]MM] [[[[]MM]]MM] [MMM] [[[[]MM]M]MM] [[MM]MM] [[[]MM]MM] [MMM] [][MMM] [MMM][MMM] [[]MMM] [MMMM] [[[]M]] [][[[]M]] [][][[[]M]] [[]][[[]M]] [[[[]M]]][[[]M]] [[][[[]M]]][[[]M]] M[[[]M]] [[[]M]][[[]M]] [][[[]M]][[[]M]] [[[]M]][[[]M]][[[]M]] [[][[]M]] [][[][[]M]] [[[]M]][[][[]M]] [[][[]M]][[][[]M]] [[][][[]M]] [[[]M]] [[[[[]M]]][[]M]] [[[[][[]M]]][[]M]] [M[[]M]] [[[[]M]][[]M]] [[[][[]M]][[]M]] [M[[]M]] [][M[[]M]] [M[[]M]][M[[]M]] [[]M[[]M]] [MM[[]M]] [[[]M][[]M]] [][[[]M][[]M]] [[[]M][[]M]][[[]M][[]M]] [[][[]M][[]M]] [M[[]M][[]M]] [[[]M][[]M][[]M]] [[[][]M]] [][[[][]M]] [[[][]M]][[[][]M]] [[][[][]M]] [M[[][]M]] [[[]M][[][]M]] [[[][]M][[][]M]] [[[][][]M]] [[M]] [][[M]] [[]][[M]] [[M]][[M]] [[][M]] [[M][M]] [[[][[]]M]] [[[[]]M]] [[[[][]]M]] [[[]M]] [[[M]M]] [[[[[[]M]]]M]] [][[[[[[]M]]]M]] [[]][[[[[[]M]]]M]] [[M]][[[[[[]M]]]M]] [[[[[[]M]]]M]][[[[[[]M]]]M]] [[][[[[[]M]]]M]] [[M][[[[[]M]]]M]] [[[[[[]M]]]M][[[[[]M]]]M]] [[[][[[[]M]]]M]] [[[[[[]M]]]M]] [[[[[[]M]]][[[[]M]]]M]] [[[[][[[]M]]]M]] [[[[[[]M]][[[]M]]]M]] [[[[[][[]M]]]M]] [[[[M[[]M]]]M]] [[[[[[]M][[]M]]]M]] [[[[[[][]M]]]M]] [[[[[M]]]M]] [[[[[[[[[]M]]]M]]]M]] [[MM]] [][[MM]] [[[]M]][[MM]] [[MM]][[MM]] [[][MM]] [M[MM]] [[[]M][MM]] [[MM][MM]] [[[]MM]] [[MMM]] [[[[]M]M]] [[[[[[]M]]M]M]] [[[MM]M]] [[[[[]M]]M]] [][[[[[]M]]M]] [[[]M]][[[[[]M]]M]] [[[[[]M]]M]][[[[[]M]]M]] [[][[[[]M]]M]] [M[[[[]M]]M]] [[[]M][[[[]M]]M]] [[[[[]M]]M][[[[]M]]M]] [[[][[[]M]]M]] [[[[[]M]][[[]M]]M]] [[[[][[]M]]M]] [[[[[[]M]][[]M]]M]] [[[M[[]M]]M]] [[[[[]M][[]M]]M]] [[[[[][]M]]M]] [[[[M]]M]] [[[[[[[[]M]]]M]]M]] [[[[[[[[[[]M]]M]]]M]]M]] [[[[MM]]M]] [[[[[[[]M]]M]]M]] [[MM]] [][[MM]] [][][[MM]] [[]][[MM]] [[[MM]]][[MM]] [[][[MM]]][[MM]] M[[MM]] [[[]M]][[MM]] [[MM]][[MM]] [][[MM]][[MM]] [[MM]][[MM]][[MM]] [[][MM]] [][[][MM]] [[MM]][[][MM]] [[][MM]][[][MM]] [[][][MM]] [[MM]] [[[[[][MM]]M]][MM]] [[[MM]][MM]] [[[][MM]][MM]] [M[MM]] [][M[MM]] [M[MM]][M[MM]] [[]M[MM]] [MM[MM]] [[[]M][MM]] [][[[]M][MM]] [[[]M][MM]][[[]M][MM]] [[][[]M][MM]] [M[[]M][MM]] [[[]M][[]M][MM]] [[[][]M][MM]] [[M][MM]] [[[[[[]M][MM]]]M][MM]] [[MM][MM]] [[[[[]M]]M][MM]] [[[[[[[]M][MM]]M]]M][MM]] [[[[MM]]M][MM]] [[[[[]M][MM]]M][MM]] [[MM][MM]] [][[MM][MM]] [[MM][MM]][[MM][MM]] [[][MM][MM]] [M[MM][MM]] [[[]M][MM][MM]] [[MM][MM][MM]] [[[]MM]] [][[[]MM]] [[[]MM]][[[]MM]] [[][[]MM]] [M[[]MM]] [[[]M][[]MM]] [[MM][[]MM]] [[[]MM][[]MM]] [[[][]MM]] [[MM]] [[[[[[]MM]]]MM]] [[MMM]] [[[[[]M]]MM]] [[[[[[[]MM]]M]]MM]] [[[[MM]]MM]] [[[[[]MM]]MM]] [[MMM]] [][[MMM]] [[MMM]][[MMM]] [[][MMM]] [M[MMM]] [[[]M][MMM]] [[MM][MMM]] [[[]MM][MMM]] [[MMM][MMM]] [[[]MMM]] [[MMMM]] [[[[]M]M]] [][[[[]M]M]] [[]][[[[]M]M]] [[[[]M]M]][[[[]M]M]] [[][[[]M]M]] [[[[]M]M][[[]M]M]] [[[][[]M]M]] [[M[[]M]M]] [[[[]M][[]M]M]] [[[[][]M]M]] [[[M]M]] [[[[[[[]M]M]]M]M]] [[[MM]M]] [][[[MM]M]] [[[MM]M]][[[MM]M]] [[][[MM]M]] [[[[]M]M][[MM]M]] [[[MM]M][[MM]M]] [[[][MM]M]] [[M[MM]M]] [[[[]M][MM]M]] [[[MM][MM]M]] [[[[]MM]M]] [[[MMM]M]] [[[[[]M]M]M]] [[[[MM]M]M]] L Total 353 ordinals. With additional fs elements [] [][] [][][] [][][][] [][][][][] [][][][][][] [][][][][][][] [][][][][][][][] [][][][][][][][][] [[]] [][[]] [][][[]] [][][][[]] [][][][][[]] [][][][][][[]] [][][][][][][[]] [][][][][][][][[]] [[]][[]] [][[]][[]] [][][[]][[]] [][][][[]][[]] [][][][][[]][[]] [][][][][][[]][[]] [][][][][][][[]][[]] [[]][[]][[]] [[]][[]][[]][[]] [[]][[]][[]][[]][[]] [[]][[]][[]][[]][[]][[]] [[]][[]][[]][[]][[]][[]][[]] [[]][[]][[]][[]][[]][[]][[]][[]] [[][]] [][[][]] [][][[][]] [][][][[][]] [][][][][[][]] [][][][][][[][]] [][][][][][][[][]] [[]][[][]] [[]][[]][[][]] [[]][[]][[]][[][]] [[]][[]][[]][[]][[][]] [[]][[]][[]][[]][[]][[][]] [[]][[]][[]][[]][[]][[]][[][]] [[][]][[][]] [[][]][[][]][[][]] [[][]][[][]][[][]][[][]] [[][]][[][]][[][]][[][]][[][]] [[][]][[][]][[][]][[][]][[][]][[][]] [[][]][[][]][[][]][[][]][[][]][[][]][[][]] [[][][]] [[][][][]] [[][][][][]] [[][][][][][]] [[][][][][][][]] [[][][][][][][][]] [] [][] [][] [[]][] [[][]][] [[][][]][] [[][][][]][] [][] [][][] [][][][] [][][][][] [][][][][][] [][][][][][][] [[][[]]] [[][][[]]] [[][][][[]]] [[][][][][[]]] [[][][][][][[]]] [[][][][][][][[]]] [[[]]] [[[]][[]]] [[[]][[]][[]]] [[[]][[]][[]][[]]] [[[]][[]][[]][[]][[]]] [[[]][[]][[]][[]][[]][[]]] [[[][]]] [[[][][]]] [[[][][][]]] [[[][][][][]]] [[[][][][][][]]] [[[][][][][][][]]] [[]] [[[]]] [[[[]]]] [[[[[]]]]] [[[[[[]]]]]] [[[[[[[]]]]]]] M []M [][]M [][][]M [][][][]M [][][][][]M [][][][][][]M [][][][][][][]M [][][][][][][][]M [[]]M [][[]]M [][][[]]M [][][][[]]M [][][][][[]]M [][][][][][[]]M [][][][][][][[]]M [[]][[]]M [[]][[]][[]]M [[]][[]][[]][[]]M [[]][[]][[]][[]][[]]M [[]][[]][[]][[]][[]][[]]M [[]][[]][[]][[]][[]][[]][[]]M [[][]]M [[][][]]M [[][][][]]M [[][][][][]]M [[][][][][][]]M [[][][][][][][]]M []M [[]]M [[[]]]M [[[[]]]]M [[[[[]]]]]M [[[[[[]]]]]]M [M]M [][M]M [[]][M]M [][M]M [[]][M]M [[[]]][M]M [[[[]]]][M]M [M][M]M [M][M][M]M [M][M][M][M]M [M][M][M][M][M]M [M][M][M][M][M][M]M [M][M][M][M][M][M][M]M [[]M]M [][[]M]M [[]][[]M]M [][[]M]M 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[[[]MM]][[[]MM]][[[]MM]][[[]MM]][[[]MM]][[[]MM]] [[[]MM]][[[]MM]][[[]MM]][[[]MM]][[[]MM]][[[]MM]][[[]MM]] [[][[]MM]] [[[][[]MM]][[]MM]] [[[[][[]MM]][[]MM]][[]MM]] [[[[[][[]MM]][[]MM]][[]MM]][[]MM]] [[[[[[][[]MM]][[]MM]][[]MM]][[]MM]][[]MM]] [[[[[[[][[]MM]][[]MM]][[]MM]][[]MM]][[]MM]][[]MM]] [M[[]MM]] [MM[[]MM]] [MMM[[]MM]] [MMMM[[]MM]] [MMMMM[[]MM]] [MMMMMM[[]MM]] [[[]M][[]MM]] [[[[[]M][[]MM]]M][[]MM]] [[[[[[[]M][[]MM]]M][[]MM]]M][[]MM]] [[[[[[[[[]M][[]MM]]M][[]MM]]M][[]MM]]M][[]MM]] [[[[[[[[[[[]M][[]MM]]M][[]MM]]M][[]MM]]M][[]MM]]M][[]MM]] [[[[[[[[[[[[[]M][[]MM]]M][[]MM]]M][[]MM]]M][[]MM]]M][[]MM]]M][[]MM]] [[MM][[]MM]] [[MM][MM][[]MM]] [[MM][MM][MM][[]MM]] [[MM][MM][MM][MM][[]MM]] [[MM][MM][MM][MM][MM][[]MM]] [[MM][MM][MM][MM][MM][MM][[]MM]] [[[]MM][[]MM]] [[[]MM][[]MM][[]MM]] [[[]MM][[]MM][[]MM][[]MM]] [[[]MM][[]MM][[]MM][[]MM][[]MM]] [[[]MM][[]MM][[]MM][[]MM][[]MM][[]MM]] [[[]MM][[]MM][[]MM][[]MM][[]MM][[]MM][[]MM]] [[[][]MM]] [[[][][]MM]] [[[][][][]MM]] [[[][][][][]MM]] [[[][][][][][]MM]] [[[][][][][][][]MM]] [[MM]] [[MM]] [[[[[MM]]]MM]] [[[[[MM][MM]]]MM]] [[[[[MM][MM][MM]]]MM]] [[[[[MM][MM][MM][MM]]]MM]] [[[[[[]MM]]]MM]] [[[[[[[[[]MM]]]MM]]]MM]] [[[[[[[[[[[[]MM]]]MM]]]MM]]]MM]] [[[[[[[[[[[[[[[]MM]]]MM]]]MM]]]MM]]]MM]] [[[[[[[[[[[[[[[[[[]MM]]]MM]]]MM]]]MM]]]MM]]]MM]] [[[[[[[[[[[[[[[[[[[[[]MM]]]MM]]]MM]]]MM]]]MM]]]MM]]]MM]] [[MMM]] [[[MM]MM]] [[[MMM]MM]] [[[MMMM]MM]] [[[MMMMM]MM]] [[[MMMMMM]MM]] [[[[[]M]]MM]] [[[[[]M]]MM]] [[[[[[MM]]M]]MM]] [[[[[[MM][MM]]M]]MM]] [[[[[[MM][MM][MM]]M]]MM]] [[[[[[MM][MM][MM][MM]]M]]MM]] [[[[[[[]MM]]M]]MM]] [[[[[[[[[[[]MM]]M]]MM]]M]]MM]] [[[[[[[[[[[[[[[]MM]]M]]MM]]M]]MM]]M]]MM]] [[[[[[[[[[[[[[[[[[[]MM]]M]]MM]]M]]MM]]M]]MM]]M]]MM]] [[[[[[[[[[[[[[[[[[[[[[[]MM]]M]]MM]]M]]MM]]M]]MM]]M]]MM]]M]]MM]] [[[[[[[[[[[[[[[[[[[[[[[[[[[]MM]]M]]MM]]M]]MM]]M]]MM]]M]]MM]]M]]MM]]M]]MM]] [[[[MM]]MM]] [[[[MM][MM]]MM]] [[[[MM][MM][MM]]MM]] [[[[MM][MM][MM][MM]]MM]] [[[[MM][MM][MM][MM][MM]]MM]] [[[[MM][MM][MM][MM][MM][MM]]MM]] [[[[[]MM]]MM]] [[[[[[[]MM]]MM]]MM]] [[[[[[[[[]MM]]MM]]MM]]MM]] [[[[[[[[[[[]MM]]MM]]MM]]MM]]MM]] [[[[[[[[[[[[[]MM]]MM]]MM]]MM]]MM]]MM]] [[[[[[[[[[[[[[[]MM]]MM]]MM]]MM]]MM]]MM]]MM]] [[MMM]] [][[MMM]] [[[]MM]][[MMM]] [[[[[]MM]]MM]][[MMM]] [[[[[[[]MM]]MM]]MM]][[MMM]] [[[[[[[[[]MM]]MM]]MM]]MM]][[MMM]] [[[[[[[[[[[]MM]]MM]]MM]]MM]]MM]][[MMM]] [[MMM]][[MMM]] [[MMM]][[MMM]][[MMM]] [[MMM]][[MMM]][[MMM]][[MMM]] [[MMM]][[MMM]][[MMM]][[MMM]][[MMM]] [[MMM]][[MMM]][[MMM]][[MMM]][[MMM]][[MMM]] [[MMM]][[MMM]][[MMM]][[MMM]][[MMM]][[MMM]][[MMM]] [[][MMM]] [[[][MMM]][MMM]] [[[[][MMM]][MMM]][MMM]] [[[[[][MMM]][MMM]][MMM]][MMM]] [[[[[[][MMM]][MMM]][MMM]][MMM]][MMM]] [[[[[[[][MMM]][MMM]][MMM]][MMM]][MMM]][MMM]] [M[MMM]] [MM[MMM]] [MMM[MMM]] [MMMM[MMM]] [MMMMM[MMM]] [MMMMMM[MMM]] [[[]M][MMM]] [[[[[]M][MMM]]M][MMM]] [[[[[[[]M][MMM]]M][MMM]]M][MMM]] [[[[[[[[[]M][MMM]]M][MMM]]M][MMM]]M][MMM]] [[[[[[[[[[[]M][MMM]]M][MMM]]M][MMM]]M][MMM]]M][MMM]] [[[[[[[[[[[[[]M][MMM]]M][MMM]]M][MMM]]M][MMM]]M][MMM]]M][MMM]] [[MM][MMM]] [[MM][MM][MMM]] [[MM][MM][MM][MMM]] [[MM][MM][MM][MM][MMM]] [[MM][MM][MM][MM][MM][MMM]] [[MM][MM][MM][MM][MM][MM][MMM]] [[[]MM][MMM]] [[[[[]MM][MMM]]MM][MMM]] [[[[[[[]MM][MMM]]MM][MMM]]MM][MMM]] [[[[[[[[[]MM][MMM]]MM][MMM]]MM][MMM]]MM][MMM]] [[[[[[[[[[[]MM][MMM]]MM][MMM]]MM][MMM]]MM][MMM]]MM][MMM]] [[[[[[[[[[[[[]MM][MMM]]MM][MMM]]MM][MMM]]MM][MMM]]MM][MMM]]MM][MMM]] [[MMM][MMM]] [[MMM][MMM][MMM]] [[MMM][MMM][MMM][MMM]] [[MMM][MMM][MMM][MMM][MMM]] [[MMM][MMM][MMM][MMM][MMM][MMM]] [[MMM][MMM][MMM][MMM][MMM][MMM][MMM]] [[[]MMM]] [[[[[]MMM]]MMM]] [[[[[[[]MMM]]MMM]]MMM]] [[[[[[[[[]MMM]]MMM]]MMM]]MMM]] [[[[[[[[[[[]MMM]]MMM]]MMM]]MMM]]MMM]] [[[[[[[[[[[[[]MMM]]MMM]]MMM]]MMM]]MMM]]MMM]] [[MMMM]] [[MMMMM]] [[MMMMMM]] [[MMMMMMM]] [[MMMMMMMM]] [[MMMMMMMMM]] [[[[]M]M]] [][[[[]M]M]] [][][[[[]M]M]] [][][][[[[]M]M]] [][][][][[[[]M]M]] [][][][][][[[[]M]M]] [][][][][][][[[[]M]M]] [[]][[[[]M]M]] M[[[[]M]M]] [[MM]][[[[]M]M]] [[MMM]][[[[]M]M]] [[MMMM]][[[[]M]M]] [[MMMMM]][[[[]M]M]] [[[[]M]M]][[[[]M]M]] [[[[]M]M]][[[[]M]M]][[[[]M]M]] [[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]] [[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]] [[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]] [[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]][[[[]M]M]] [[][[[]M]M]] [M[[[]M]M]] [[MM][[[]M]M]] [[MMM][[[]M]M]] [[MMMM][[[]M]M]] [[MMMMM][[[]M]M]] [[[[]M]M][[[]M]M]] [[[[]M]M][[[]M]M][[[]M]M]] [[[[]M]M][[[]M]M][[[]M]M][[[]M]M]] [[[[]M]M][[[]M]M][[[]M]M][[[]M]M][[[]M]M]] [[[[]M]M][[[]M]M][[[]M]M][[[]M]M][[[]M]M][[[]M]M]] [[[[]M]M][[[]M]M][[[]M]M][[[]M]M][[[]M]M][[[]M]M][[[]M]M]] [[[][[]M]M]] [[[[[][[]M]M]][[]M]M]] [[[[[[[][[]M]M]][[]M]M]][[]M]M]] [[[[[[[[[][[]M]M]][[]M]M]][[]M]M]][[]M]M]] [[[[[[[[[[[][[]M]M]][[]M]M]][[]M]M]][[]M]M]][[]M]M]] [[[[[[[[[[[[[][[]M]M]][[]M]M]][[]M]M]][[]M]M]][[]M]M]][[]M]M]] [[M[[]M]M]] [[MM[[]M]M]] [[MMM[[]M]M]] [[MMMM[[]M]M]] [[MMMMM[[]M]M]] [[MMMMMM[[]M]M]] [[[[]M][[]M]M]] [[[[]M][[]M][[]M]M]] [[[[]M][[]M][[]M][[]M]M]] [[[[]M][[]M][[]M][[]M][[]M]M]] [[[[]M][[]M][[]M][[]M][[]M][[]M]M]] [[[[]M][[]M][[]M][[]M][[]M][[]M][[]M]M]] [[[[][]M]M]] [[[[][][]M]M]] [[[[][][][]M]M]] [[[[][][][][]M]M]] [[[[][][][][][]M]M]] [[[[][][][][][][]M]M]] [[[M]M]] [[[MM]M]] [[[[[MM]]M]M]] [[[[[MMM]]M]M]] [[[[[MMMM]]M]M]] [[[[[MMMMM]]M]M]] [[[[[[[]M]M]]M]M]] [[[[[[[[[[]M]M]]M]M]]M]M]] [[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]] [[[[[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]]M]M]] [[[[[[[[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]]M]M]]M]M]] [[[[[[[[[[[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]]M]M]]M]M]]M]M]] [[[MM]M]] [][[[MM]M]] [[[[]M]M]][[[MM]M]] [[[[[[[]M]M]]M]M]][[[MM]M]] [[[[[[[[[[]M]M]]M]M]]M]M]][[[MM]M]] [[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]][[[MM]M]] [[[[[[[[[[[[[[[[]M]M]]M]M]]M]M]]M]M]]M]M]][[[MM]M]] [[[MM]M]][[[MM]M]] [[[MM]M]][[[MM]M]][[[MM]M]] [[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]] [[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]] [[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]] [[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]][[[MM]M]] [[][[MM]M]] [M[[MM]M]] [[MM][[MM]M]] [[MMM][[MM]M]] [[MMMM][[MM]M]] [[MMMMM][[MM]M]] [[[[]M]M][[MM]M]] [[[[[[[]M]M][[MM]M]]M]M][[MM]M]] [[[[[[[[[[]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]] [[[[[[[[[[[[[]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]] [[[[[[[[[[[[[[[[]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]] [[[[[[[[[[[[[[[[[[[]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]]M]M][[MM]M]] [[[MM]M][[MM]M]] [[[MM]M][[MM]M][[MM]M]] [[[MM]M][[MM]M][[MM]M][[MM]M]] [[[MM]M][[MM]M][[MM]M][[MM]M][[MM]M]] [[[MM]M][[MM]M][[MM]M][[MM]M][[MM]M][[MM]M]] [[[MM]M][[MM]M][[MM]M][[MM]M][[MM]M][[MM]M][[MM]M]] [[[][MM]M]] [[[[[][MM]M]][MM]M]] [[[[[[[][MM]M]][MM]M]][MM]M]] [[[[[[[[[][MM]M]][MM]M]][MM]M]][MM]M]] [[[[[[[[[[[][MM]M]][MM]M]][MM]M]][MM]M]][MM]M]] [[[[[[[[[[[[[][MM]M]][MM]M]][MM]M]][MM]M]][MM]M]][MM]M]] [[M[MM]M]] [[MM[MM]M]] [[MMM[MM]M]] [[MMMM[MM]M]] [[MMMMM[MM]M]] [[MMMMMM[MM]M]] [[[[]M][MM]M]] [[[[[[[]M][MM]M]]M][MM]M]] [[[[[[[[[[]M][MM]M]]M][MM]M]]M][MM]M]] [[[[[[[[[[[[[]M][MM]M]]M][MM]M]]M][MM]M]]M][MM]M]] [[[[[[[[[[[[[[[[]M][MM]M]]M][MM]M]]M][MM]M]]M][MM]M]]M][MM]M]] [[[[[[[[[[[[[[[[[[[]M][MM]M]]M][MM]M]]M][MM]M]]M][MM]M]]M][MM]M]]M][MM]M]] [[[MM][MM]M]] [[[MM][MM][MM]M]] [[[MM][MM][MM][MM]M]] [[[MM][MM][MM][MM][MM]M]] [[[MM][MM][MM][MM][MM][MM]M]] [[[MM][MM][MM][MM][MM][MM][MM]M]] [[[[]MM]M]] [[[[[[[]MM]M]]MM]M]] [[[[[[[[[[]MM]M]]MM]M]]MM]M]] [[[[[[[[[[[[[]MM]M]]MM]M]]MM]M]]MM]M]] [[[[[[[[[[[[[[[[]MM]M]]MM]M]]MM]M]]MM]M]]MM]M]] [[[[[[[[[[[[[[[[[[[]MM]M]]MM]M]]MM]M]]MM]M]]MM]M]]MM]M]] [[[MMM]M]] [[[MMMM]M]] [[[MMMMM]M]] [[[MMMMMM]M]] [[[MMMMMMM]M]] [[[MMMMMMMM]M]] [[[[[]M]M]M]] [[[[[[[[[]M]M]M]]M]M]M]] [[[[[[[[[[[[[]M]M]M]]M]M]M]]M]M]M]] [[[[[[[[[[[[[[[[[]M]M]M]]M]M]M]]M]M]M]]M]M]M]] [[[[[[[[[[[[[[[[[[[[[]M]M]M]]M]M]M]]M]M]M]]M]M]M]]M]M]M]] [[[[[[[[[[[[[[[[[[[[[[[[[]M]M]M]]M]M]M]]M]M]M]]M]M]M]]M]M]M]]M]M]M]] [[[[MM]M]M]] [[[[[MM]M]M]M]] [[[[[[MM]M]M]M]M]] [[[[[[[MM]M]M]M]M]M]] [[[[[[[[MM]M]M]M]M]M]M]] [[[[[[[[[MM]M]M]M]M]M]M]M]] L Total 1838 ordinals. Pentuple expansion Clean [] [][] [][][] [][][][] [][][][][] [[]] [][[]] [][][[]] [][][][[]] [[]][[]] [][[]][[]] [][][[]][[]] [[]][[]][[]] [][[]][[]][[]] [[]][[]][[]][[]] [[][]] [][[][]] [][][[][]] [[]][[][]] [][[]][[][]] [[]][[]][[][]] [[][]][[][]] [][[][]][[][]] [[]][[][]][[][]] [[][]][[][]][[][]] [[][][]] [][[][][]] [[]][[][][]] [[][]][[][][]] [[][][]][[][][]] [[][][][]] [] [][] [][][] [[]][] [][] [][][] [][][] [[][[]]] [][[][[]]] [][[][[]]] [[][[]]][[][[]]] [[][][[]]] [[[]]] [][[[]]] [[[]]][[[]]] [[][[]][[]]] [[[]][[]]] [[[][]]] [][[[][]]] [[[][]]][[[][]]] [[][[][]]] [[[][]]] [[[][]][[][]]] [[[][][]]] [[]] [][[]] [[]][[]] [[]][[]] [[][]] [[][]] [[[][[]]]] [[[[]]]] [[[[][]]]] [[[]]] M []M [][]M [][][]M [][][][]M [[]]M [][[]]M [][][[]]M [[]][[]]M [][[]][[]]M [[]][[]][[]]M [[][]]M [][[][]]M [[]][[][]]M [[][]][[][]]M [[][][]]M []M [][]M [][]M [[][[]]]M [[[]]]M [[[][]]]M [[]]M [M]M [][M]M [][][M]M [[]][M]M [M][M]M [][M][M]M [M][M][M]M [[]M]M [][[]M]M [][][[]M]M [[]][[]M]M [M][[]M]M 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[[[[]M][MM]M]][[[[]M][MM]M]] [[][[[]M][MM]M]] [[[[]M][MM]M][[[]M][MM]M]] [[[][[]M][MM]M]] [[M[[]M][MM]M]] [[[[]M][[]M][MM]M]] [[[[][]M][MM]M]] [[[M][MM]M]] [[[[[[[]M][MM]M]]M][MM]M]] [[[MM][MM]M]] [][[[MM][MM]M]] [[[MM][MM]M]][[[MM][MM]M]] [[][[MM][MM]M]] [[[[]M][MM]M][[MM][MM]M]] [[[MM][MM]M][[MM][MM]M]] [[[][MM][MM]M]] [[M[MM][MM]M]] [[[[]M][MM][MM]M]] [[[MM][MM][MM]M]] [[[[]MM]M]] [][[[[]MM]M]] [[]][[[[]MM]M]] [[[[]MM]M]][[[[]MM]M]] [[][[[]MM]M]] [[[[]MM]M][[[]MM]M]] [[[][[]MM]M]] [[M[[]MM]M]] [[[[]M][[]MM]M]] [[[MM][[]MM]M]] [[[[]MM][[]MM]M]] [[[[][]MM]M]] [[[MM]M]] [[[[[[[]MM]M]]MM]M]] [[[MMM]M]] [][[[MMM]M]] [[[MMM]M]][[[MMM]M]] [[][[MMM]M]] [[[[]MM]M][[MMM]M]] [[[MMM]M][[MMM]M]] [[[][MMM]M]] [[M[MMM]M]] [[[[]M][MMM]M]] [[[MM][MMM]M]] [[[[]MM][MMM]M]] [[[MMM][MMM]M]] [[[[]MMM]M]] [[[MMMM]M]] [[[[[]M]M]M]] [][[[[[]M]M]M]] [[[]M]][[[[[]M]M]M]] [[[[[]M]M]M]][[[[[]M]M]M]] [[][[[[]M]M]M]] [M[[[[]M]M]M]] [[[]M][[[[]M]M]M]] [[[[[]M]M]M][[[[]M]M]M]] [[[][[[]M]M]M]] [[[[[]M]M][[[]M]M]M]] [[[[][[]M]M]M]] [[[M[[]M]M]M]] [[[[[]M][[]M]M]M]] [[[[[][]M]M]M]] [[[[M]M]M]] [[[[[[[[]M]]]M]M]M]] [[[[[[[[[[]M]M]M]]]M]M]M]] [[[[MM]M]M]] [[[[[[[]M]]M]M]M]] [[[[[[[[[]M]M]M]]M]M]M]] [[[[MM]M]M]] [][[[[MM]M]M]] [[[[MM]M]M]][[[[MM]M]M]] [[][[[MM]M]M]] [M[[[MM]M]M]] [[[]M][[[MM]M]M]] [[[[[]M]M]M][[[MM]M]M]] [[[[MM]M]M][[[MM]M]M]] [[[][[MM]M]M]] [[[[[]M]M][[MM]M]M]] [[[[MM]M][[MM]M]M]] [[[[][MM]M]M]] [[[M[MM]M]M]] [[[[[]M][MM]M]M]] [[[[MM][MM]M]M]] [[[[[]MM]M]M]] [[[[MMM]M]M]] [[[[[[]M]M]M]M]] [[[[[MM]M]M]M]] L Total 2833 ordinals. 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